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Gareth Roberts studies Markov chain Monte Carlo and the mathematical efficiency of Bayesian computation. With Giacomo Zanella, he developed importance tempering for high-dimensional sampling.^[ref]

Their Tempered Gibbs sampler changes how coordinates are selected. Its invariant law differs from the intended target, so importance sampling weights correct the averages. This trades easier movement through the state space against the variability of the correcting weights, with explicit theoretical comparisons to ordinary sampling methods.^[ref]

Giacomo Zanella develops methods in computational statistics, applied probability theory and machine learning, especially Bayesian computation. His research statement identifies Markov chain Monte Carlo, random measures and variable selection among these interests.^[ref]

With Gareth Roberts, he proposed the Tempered Gibbs sampler. A modified coordinate-selection rule can explore difficult high-dimensional targets more effectively; importance sampling weights then recover averages under the desired probability distribution. Their analysis includes applications to Bayesian variable selection.^[ref]

Tempered Gibbs sampling combines modified full conditional distributions with state-dependent coordinate selection and importance sampling. For a positive target density $f(x)$, choose strictly positive normalized densities $g_i(x_i\mid x_{-i})$ with compatible support, and defineSelect coordinate $i$ proportionally to $p_i(x)$ and redraw it from $g_i$. The invariant density is $f(x)Z(x)$, not generally $f(x)$. For a move changing only coordinate $i$, stationary transition weights are proportional to $f(x_{-i})g_i(x_i\mid x_{-i})g_i(y_i\mid x_{-i})$, symmetric in old and new values. Integrating also shows $\int fZ=1$.

Under the appropriate Markov-chain law of large numbers, weighted averages with weights $1/Z(x)$ recover expectations under $f$. Flattening a conditional can encourage longer moves, but omitting the weights changes the estimand. The $g_i$ need not be full conditionals of a common second joint distribution.^[ref]